Are standing osmotic gradients the main driver of cerebrospinal fluid production? A computational analysis

Background The mechanisms of cerebrospinal fluid (CSF) production by the ventricular choroid plexus (ChP) have not been fully deciphered. One prominent hypothesized mechanism is trans-epithelial water transport mediated by accumulation of solutes at the luminal ChP membrane that produces local osmotic gradients. However, this standing osmotic gradient hypothesis has not been systematically tested. Methods To assess the plausibility of the standing gradient mechanism serving as the main driver of CSF production by the ChP, we developed a three-dimensional (3D) and a one-dimensional (1D) computational model to quantitatively describe the associated processes in the rat ChP inter-microvillar spaces and in CSF pools between macroscopic ChP folds (1D only). The computationally expensive 3D model was used to examine the applicability of the 1D model for hypothesis testing. The 1D model was employed to predict the rate of CSF produced by the standing gradient mechanism for 200,000 parameter permutations. Model parameter values for each permutation were chosen by random sampling from distributions derived from published experimental data. Results Both models predict that the CSF production rate by the standing osmotic gradient mechanism is below 10% of experimentally measured values that reflect the contribution of all actual production mechanisms. The 1D model indicates that increasing the size of CSF pools between ChP folds, where diffusion dominates solute transport, would increase the contribution of the standing gradient mechanism to CSF production. Conclusions The models suggest that the effect of standing osmotic gradients is too small to contribute substantially to CSF production. ChP motion and movement of CSF in the ventricles, which are not accounted for in the models, would further reduce this effect, making it unlikely that standing osmotic gradients are the main drivers of CSF production. Supplementary Information The online version contains supplementary material available at 10.1186/s12987-023-00419-2.


Additional File 1 Derivation of the FU Tip Boundary Condition
The protected zone is an open area (without any membrane) where solute transport is governed by the advection-diffusion equation (S1). Note that there is no CSF production in the protected zone, but uniform velocity results from CSF entering from the FU. The general form of the solute concentration profile in the protected region is given by Equation (S2), where 1 , 2 , and are unknown. The continuity of the concentration profile everywhere necessitates equal concentration value ( ) and slope (d /d ) at the interface of the FU and the protected region ( = mv ). At the ventricular end of the protected region, the concentration is equal to the bulk CSF concentration (Equation (S3)), and at its FU end, the value and the slope are equal to the corresponding ones inside the FU (Equations (S4) and (S5)). Subtracting Equation (S3) from (S4) and substituting 2 into Equation (S5) yields Equation (8).

Estimation of Choroid Plexus Surface Area and Luminal Membrane Permeability
The apparent area of the ChP, app , cannot be measured easily because of its convoluted surface, but it can be estimated from its weight as app = .
. ℎ where, , , ℎ, and e are the ChP epithelial ratio, ChP mass, cell height, and the tissue density. The value distribution of these parameters is shown in Table S1. The luminal membrane permeability, p , can also be derived from other parameters as Here, Δ is the transepithelial concentration difference (between blood and CSF in the ventricular space). Note that the epithelial resistance to water transport is assumed to be equally divided between luminal and basolateral membranes.   Table 2. (b) A schematic distribution of concentration along a typical FU and protected zone with parametric representation of the main points.

Péclet Number Calculation
The Péclet number ( ) is a measure for the relevance of convection compared to diffusion for solute transport. Figure S1a shows that the Péclet number, defined as = | = mv ⋅ prot / , is very small for all possible parameter permutations. As a result, solute transport in the FU and the protected region is dominated by diffusion, and the small approximation for Equations (6) -(8) can be applied. In this case, Equation (8) (6) and (7) yields Equation (S9). Combining Equations (S8) and (S9) results in Equation (S10), which shows a linear relation between concentration drop along the protected region and prot .
Here, 1 and 1 are, respectively, the velocity and concentration on the interface between FU and the protected region (Fig. S1b). The production rate p is calculated from which can be decomposed into two terms: i.e., the second term in Equation (S12) has a much stronger contribution to the production rate (see Fig. S1b) and the first one can be neglected. Here, 2 denotes the concentration at the base of the FU. Thus, the production rate is approximately linearly related to the protected length as well: To assess the validity of the approximated model, Equation (S14) was used to calculate the CSF production rate for the same parameter permutations used in the full 1D model. Figure   S2a shows that for parameter permutations that result in small CSF production rates, predictions of the full and approximated 1D model match well. At higher production rates, the approximated model overpredicts CSF production (Fig. S2b).

Fig. S2.
Comparison of the full 1D model results (in black) with those of its small approximation (in red). (a) Probability density of the predicted CSF production rate using the same parameter permutations. (b) CSF production rates predicted by the full 1D model plotted against those of its approximation. Perfect correlation between the two would follow the black dashed line.

Estimation of the Protected Length
As shown in the magnified view in Fig. 3c, the folds of ChP consist of two regions: the intermicrovillar space and the protected region. CSF pools of the protected region are separated from the bulk CSF flow by a virtual surface enveloping the ChP. The exact distribution of the protected length could be determined precisely from the shape of the luminal ChP surface and the envelope surface (see Fig. 3d), but this would require currently unavailable in vivo high-resolution scans of the entire rat ChP. As a workaround, the average protected length can be estimated from available data.
The average protected length is calculated as where is the total number of FUs on the ChP surface. We assume that the ChP has f folds, each one having ( ) FUs. Each FU is located at a distance prot ( ) (corresponding to the th FU in the th fold) away from bulk CSF. We note that ∑ ( ) f =1 is smaller than since some FUs are located outside of folds (corresponding to zero protected length). These FUs do not contribute to the nominator of Equation (S15). We define as the fraction of the ChP surface in contact with the envelope surface (where the protected length is zero). Then, 1 − is the fraction of the ChP surface within folds, i.e., where the protected length is greater than zero.
For a graphical interpretation of λ, Fig. S3 shows three artificial ChP morphologies for which the fraction of ChP in contact with the envelope varies from 0 to 1. We use ChP to designate the area of a ChP surface patch covering one FU, and env the area of the juxtaposed envelope patch. Therefore, for each fold, the number of envelope and -6-surface patches is equal (see Fig. 3 in the paper). The value of ChP is the same for all FUs (since uniform FUs were used in the models) and thus for all folds. env is chosen to be constant within each fold. Multiplying both the nominator and the denominator of Equation (S15) with ChP , we obtain The term ( ChP / env ( ) ) is 1 where the ChP surface and its envelope coincide, and >1 in the folds, since there the ChP has to cover a larger area than the envelope. This ratio may be different for each fold, since the envelop patch area depends on the fold shape. We now geometrically rearrange the individual folds to merge them into one large fold without changing the protected lengths of the individual FUs. This is permissible since maintaining the protected lengths ensures that the CSF production does not change. Also, it does not change the average protected length according to Equation (S16). The envelope area of the combined single fold is made up of patches of different sizes. We now scale the individual envelope patch areas to one average area, env ̅̅̅̅̅̅ , and reformulate Equation (S16) accordingly: Using the globally constant ChP surface patch area and the average envelope patch area, we can now relate the patch area ratio to global ChP parameters: The numerator on the right-hand side corresponds to the ChP surface area in the fold, while the denominator is the area of the envelope covering the fold. Inserting Equation (S18) in Equation (S17), we obtain a new expression for the average protected length: Since env ̅̅̅̅̅̅ is smaller than or equal to ChP , the term prot ( ) • env ̅̅̅̅̅̅ is always smaller than the unit volume of the protected region (the blue rectangles shown in Fig. 3c). Hence, the expression in brackets in Equation (S19) is always smaller than the volume of the protected region. By replacing the expression with the total volume of protected regions, which is equal to the volume enclosed by the envelope, env , minus the total volume of the ChP, ChP , we obtain an upper limit for the average protected length: The shape of the ChP is partially accounted for by λ. For example, if the ChP does not fold ( Fig. S3c), λ is equal to one and the right-hand side of Equation (S20) becomes zero, indicating that the average protected length is zero.
To compute Equation (S20) for the actual ChP geometry, the envelope surface area is required. env cannot be larger than the inner surface area of the ventricles, vent . In the hypothetical situation where a) = 0, and b) the ChP fills the entire ventricular space, i.e., env is equal to the ventricular volume, vent , and env = vent , the protected CSF pools reach their largest possible size. Therefore, Using the Sprague Dawley rat brain atlas (v4, RRID: SCR_017124) [25], we measured the total volume and area of the ventricular space to be 34.1 mm 3 and 380 mm 2 , respectively. The ChP volume is 4.69 mm 3 (Table S1). Thus, based on Equation (S21), the average protected length is ≤ 77.5 µm. We selected this upper limit of 77.5 µm as the center value of a normal distribution of the average protected length. Therefore, the prot model parameter space extends to 155 µm, which is favorable for CSF production by the SG mechanism.